A321897 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

1, 1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 6, 3, 8, 6, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 24, 30, 20, 15, 20, 10, 1, 0, 0, 0, 1, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 0, 1, 720, 840, 504, 420, 630

Offset

1,7

Comments

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Table of n, a(n) for n=1..77.

Wikipedia, Symmetric polynomial

Example

Triangle begins:
    1
    1
    1    1
    0    1
    2    3    1
    0    1    1
    6    3    8    6    1
    0    0    1
    0    1    0    2    1
    0    0    2    3    1
   24   30   20   15   20   10    1
    0    0    0    1    1
  120   90  144   40   15   90  120   45   40   15    1
    0    6    0    3    8    6    1
    0    0    2    3    2    4    1
    0    0    0    0    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    0    0    1    0    2    1
For example, row 14 gives: 12h(41) = 6p(41) + 3p(221) + 8p(311) + 6p(2111) + p(11111).

Crossrefs

Row sums are A321898.

Cf. A005651, A008480, A056239, A124794, A124795, A319193, A321742-A321765, A321896.

Sequence in context: A079343 A004566 A321896 * A050074 A346688 A278313

Adjacent sequences: A321894 A321895 A321896 * A321898 A321899 A321900

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved